Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of the trigonometric functions for the acute angle \(\theta\). $$\sec \theta=\frac{6}{5}$$

Step-by-Step Solution

Answer

\( \cos \theta = \frac{5}{6} \), \( \sin \theta = \frac{\sqrt{11}}{6} \), \( \tan \theta = \frac{\sqrt{11}}{5} \).

Step 1: Relate Secant to Cosine

The secant function is the reciprocal of the cosine function. Therefore, if \( \sec \theta = \frac{6}{5} \), then \( \cos \theta = \frac{5}{6} \).

Step 2: Identify the Right Triangle Relationships

We know that \( \cos \theta = \frac{5}{6} \), which represents the ratio of the adjacent side to the hypotenuse in a right triangle. Let's set the adjacent side as 5 and the hypotenuse as 6.

Step 3: Use the Pythagorean Theorem to Find the Opposite Side

In a right triangle, the Pythagorean Theorem states \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. Here, the adjacent side \( (a) \) is 5, and the hypotenuse \( (c) \) is 6. Substitute to find the opposite side \( (b) \): \[5^2 + b^2 = 6^2 \]\[25 + b^2 = 36 \]\[b^2 = 11 \]\[b = \sqrt{11}\]

Step 4: Determine Sine and Tangent Values

Now that we have the lengths for all sides of the triangle: adjacent (5), hypotenuse (6), and opposite (\(\sqrt{11}\)), we can find the other trigonometric functions.- \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{11}}{6} \)- \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{11}}{5} \)

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is particularly useful when exploring angles and sides of triangles, especially in the unit circle approach.
When we talk about the secant of an angle \( \theta \), we're referring to the reciprocal of the cosine function. This means:

• \( \sec \theta = \frac{1}{\cos \theta} \)

In the given problem, \( \sec \theta = \frac{6}{5} \). By understanding the nature of secant, we can easily find the value of cosine by taking the reciprocal:

• If \( \sec \theta = \frac{6}{5} \), then \( \cos \theta = \frac{5}{6} \).

This simple relationship provides a powerful tool for solving trigonometric equations and exploring geometric concepts.

Cosine Function

The cosine function is integral to understanding right triangle geometry. In a right triangle, the cosine of an angle \( \theta \) is defined as the ratio of the length of the adjacent side to the hypotenuse:

• \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)

From the solution, we find \( \cos \theta = \frac{5}{6} \). This tells us two things:

• The length of the side adjacent to \( \theta \) is 5 units.
• The length of the hypotenuse opposite \( \theta \) is 6 units.

These values enable us to apply the Pythagorean theorem and further explore other trigonometric functions like sine and tangent. It's clear that knowing the cosine value gives a significant head start in solving for other elements within a triangle.

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of geometry, especially when working with right triangles. It states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse:

• \( a^2 + b^2 = c^2 \).

In this problem, we have the adjacent side \( (a) = 5 \) and hypotenuse \( (c) = 6 \).
To find the third side (the opposite side \( b \)), we substitute these values into the theorem:

• \( 5^2 + b^2 = 6^2 \)
• \( 25 + b^2 = 36 \)
• \( b^2 = 11 \)
• \( b = \sqrt{11} \)

Using the Pythagorean Theorem allows us not just to find the missing side, but also to explore additional trigonometric functions such as sine and tangent for the angle \( \theta \). Thus, it effectively ties together various trigonometric concepts.